Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows
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| Format: | Preprint |
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2026
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| _version_ | 1866911654886768640 |
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| author | Liao, Hong-lin Wang, Xiaoming Wang, Xuping Wen, Cao |
| author_facet | Liao, Hong-lin Wang, Xiaoming Wang, Xuping Wen, Cao |
| contents | High-order adaptive time-stepping algorithms are of significant practical value and theoretical interest for accelerating long-time fluid-flow simulations and resolving complex dynamical behaviors. While several high-order implicit-explicit schemes have been proposed in the literature, their long-time stability properties remain largely unexplored. We develop a family of long-time stable implicit-explicit Runge-Kutta (IERK) methods, up to fourth-order temporal accuracy, for the two-dimensional incompressible Navier-Stokes equations in vorticity-stream function formulation. By combining a convolution-type Hölder inequality with a damping-type multistage Grönwall inequality, we establish a unified analytical framework that proves long-time stability in both the $L^2$ and $H^1$ norms. A key component of the analysis is a mathematical-induction argument that ensures stage-wise boundedness of the vorticity in the $H^δ$ norm for some $δ>0$. To the best of our knowledge, this is the first work to establish large-time stability results for high-order IERK algorithms for the two-dimensional incompressible Navier-Stokes equations. Our IERK schemes employ stiffly accurate diagonally implicit Runge-Kutta approximations for the linear diffusive term together with explicit Runge-Kutta approximations for the nonlinear advection term. By exploiting the specific structure of the Navier-Stokes model, we derive a reduced set of order conditions-requiring only 5 and 11 conditions for the third- and fourth-order methods, respectively, in contrast to the classical 6 and 18-allowing the construction of a parameterized family of efficient schemes. These IERK methods are particularly well suited for adaptive time-stepping, as they permit significantly enlarged step sizes in actual computations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows Liao, Hong-lin Wang, Xiaoming Wang, Xuping Wen, Cao Numerical Analysis 65M12, 65M50, 76B47, 76D05 High-order adaptive time-stepping algorithms are of significant practical value and theoretical interest for accelerating long-time fluid-flow simulations and resolving complex dynamical behaviors. While several high-order implicit-explicit schemes have been proposed in the literature, their long-time stability properties remain largely unexplored. We develop a family of long-time stable implicit-explicit Runge-Kutta (IERK) methods, up to fourth-order temporal accuracy, for the two-dimensional incompressible Navier-Stokes equations in vorticity-stream function formulation. By combining a convolution-type Hölder inequality with a damping-type multistage Grönwall inequality, we establish a unified analytical framework that proves long-time stability in both the $L^2$ and $H^1$ norms. A key component of the analysis is a mathematical-induction argument that ensures stage-wise boundedness of the vorticity in the $H^δ$ norm for some $δ>0$. To the best of our knowledge, this is the first work to establish large-time stability results for high-order IERK algorithms for the two-dimensional incompressible Navier-Stokes equations. Our IERK schemes employ stiffly accurate diagonally implicit Runge-Kutta approximations for the linear diffusive term together with explicit Runge-Kutta approximations for the nonlinear advection term. By exploiting the specific structure of the Navier-Stokes model, we derive a reduced set of order conditions-requiring only 5 and 11 conditions for the third- and fourth-order methods, respectively, in contrast to the classical 6 and 18-allowing the construction of a parameterized family of efficient schemes. These IERK methods are particularly well suited for adaptive time-stepping, as they permit significantly enlarged step sizes in actual computations. |
| title | Long-time stability of implicit-explicit Runge-Kutta methods for two-dimensional incompressible flows |
| topic | Numerical Analysis 65M12, 65M50, 76B47, 76D05 |
| url | https://arxiv.org/abs/2605.05645 |